We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.
|Original language||English (US)|
|Title of host publication||Multiscale, Nonlinear and Adaptive Approximation|
|Subtitle of host publication||Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday|
|Publisher||Springer Berlin Heidelberg|
|Number of pages||61|
|State||Published - Dec 1 2009|
All Science Journal Classification (ASJC) codes